bankrollfandomcom-20200214-history
Bardi's logarithmic value of money
Bardi's logarithmic value of money is a heuristic used to make choices in bankroll management. See Generalized logarithmic value of money As an example, in a nutshell, the heuristic used is that doubling the size of a bankroll is of equal value regardless of the size of the bankroll. So if one's bankroll is 10kg of gold, the pain of losing 5kg of gold cancels the joy of gaining 10kg of gold. This seemingly very conservative, pusillanimous heuristic is justified in the following discussion. Indeed it seems that if one is unwilling to lose one's entire bankroll, one's most aggressive optimal strategy is to use Bardi's logarithmic value of money heuristic. Let us assume that one has an opportunity invest any portion of one's bankroll in something that quickly yields a three times return on investment half of the time and a total loss half of the time; and let us assume that this opportunity repeats itself indefinitely at some fixed interval for the foreseeable future. How should one invest? To maximize the average amount of gold in one's bankroll (starting at 10kg of gold) one should always invest the whole bankroll. So after four investments, one time in sixteen the bankroll would have 810kg of gold and the remaining 93.75% of the time the bankroll would be wiped out. After twenty investments the bankroll would have more than 30 million tons of gold about 1 millionth of the time and the remaining 99.9999% of the time the bankroll would be wiped out. Multiplying the average money value of the bankroll by five every four investments sounds good, but is a recipe for assured bankroll annihilation. Let us reject maximization of the average amount of gold in the bankroll. Instead let us assume that there is an optimal portion of the bankroll to invest in our half-the-time-triple-your-money-forever-investment. Using Bardi's logarithmic value of money method one finds the optimal portion of the bankroll to invest is 25%. For simple investments like the one described the formula for the bankroll portion to invest is x = p - q/o, where x is the portion of the bankroll to invest, p is the probability of the investment succeeding, q is the probability of the investment being a total loss, and o is the successful investment multiplier minus one. For the given example the value of x = 0.5 - 0.5/(3 - 1) or 25%. The mathematical derivation of x = p - q/o follows: derivative d(f(x) + g(x))/dx = df/dx + dg/dx derivative d(f(x)*g(x))/dx = f*dg/dx + g*df/dx derivative d(f(g(x)) = dg/dx*df/dg precedence derivatives and functions d f g ln (natural logarithm) exponentiation ** division / multiplication * subtraction addition - + o investment multiplier for success minus one p probability of success q probability of loss N number of investments P N times probability of success Q N times probability of loss x portion of bankroll maximize with respect to x : (1 + x*o)**P * (1 - x)**Q => ((1 + x*o)**p * (1 - x)**q)**N maximize with respect to x : (1 + x*o)**p * (1 - x)**q d(1 + x*o)**p)/dx = p*o*/(1 + x*o)**q d((1 - x)**q)/dx = q*(-1)/(1 - x)**p zero derivative at inflection point p*o*((1 - x)/(1 + x*o))**q - q*((1 + x*o)/(1 - x))**p p*o*((1 - x)/(1 + x*o))**q = q*((1 + x*o)/(1 - x))**p p*o = q*(1 + x*o)/(1 - x) p*o*(1 - x) = q*(1 + x*o) p*o - p*o*x = q + q*x*o p*o = q + x*o x = (p*o - q)/o => x = e/o where e = p*o - q maximum where e > 0 NB e <= o, x <= p x = p - q/o Alternate derivation maximize with respect to x : (1 + x*o)**p * (1 - x)**q maximize with respect to x: p*ln(1 + x*o) + q*ln(1 - x) compare to p*(1 + x*o) + q*(1 - x) which is expected average bankroll size losing half your money is as painful as doubling your money is joyful derivative p*o/(1 + x*o) - q/(1-x) p*o(1 - x) - q*(1 + x*o) ...